Subgroups — C₂×#14271212 #16828099

Order 3712.

460 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90777	C₂	1	2	yes	maximal
2	#14271212	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11745167	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₉	C₄	1	4	yes	—
7	C₂×M₁₆×C₂₉	C₄	1	4	yes	—
8	C₂×M₁₆×C₂₉	C₂²	1	4	yes	—
9	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
10	C₂×#90777	—	2	2	no	—
11	C₂×#90777	—	2	4	no	—
12	C₂×#90777	C₂²	1	4	yes	—
13	C₂×#90777	C₂²	1	8	yes	—
14	C₂²×#6597	—	2	2	no	—
15	#11745167	C₂²	1	16	yes	complemented
16	C₂⁴×C₂₉	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₉	D₄	1	2	yes	—
20	C₂²×C₄×C₂₉	D₄	1	2	yes	—
21	C₂²×C₄×C₂₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
25	M₁₆×C₂₉	C₂×C₄	1	16	yes	—
26	C₂×C₈×C₂₉	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₂₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₉	C₂³	1	4	yes	—
29	#90777	—	2	8	no	—
30	#90777	—	2	4	no	—
31	#90777	C₂³	1	8	yes	—
32	C₂×#6597	—	2	8	no	—
33	C₂×#6597	—	2	4	no	—
34	C₂³×C₂₉	—	4	2	no	—
35	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
36	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₂₉	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₂₉	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
52	C₈×C₂₉	C₂²×C₄	1	8	yes	—
53	#6597	—	2	8	no	—
54	C₂×#430	—	29	1	no	maximal, Sylow
55	C₂²×C₂₉	—	2	4	no	—
56	C₂²×C₂₉	—	2	2	no	—
57	C₂²×C₂₉	—	4	2	no	—
58	C₂²×C₂₉	—	4	2	no	—
59	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
61	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
65	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
68	C₂²×M₁₆	—	29	2	no	—
69	C₂²×M₁₆	D₂₉	1	1	yes	characteristic
70	#430	—	29	8	no	—
71	C₂×#108	—	29	4	no	—
72	C₂×C₂₉	—	2	4	no	—
73	C₂×C₂₉	—	2	1	no	—
74	C₂×C₂₉	—	2	1	no	—
75	C₂×C₂₉	#430	1	2	yes	complemented
76	C₂×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
78	C₂×M₁₆	—	29	4	no	—
79	C₂×M₁₆	—	29	8	no	—
80	C₂×M₁₆	—	58	4	no	—
81	C₂×M₁₆	—	58	2	no	—
82	C₂×M₁₆	Dic₂₉	1	4	yes	—
83	C₂×M₁₆	Dic₂₉	1	4	yes	—
84	C₂×M₁₆	C₂×D₂₉	1	4	yes	—
85	C₂²×C₈	—	58	2	no	—
86	C₂²×C₈	C₂×D₂₉	1	2	yes	—
87	#108	—	29	16	no	—
88	C₂₉	C₂×#430	1	1	yes	characteristic, complemented, Sylow
89	C₂⁴	Dic₅₈	1	1	yes	characteristic
90	C₂²×C₄	Dic₅₈	1	1	yes	characteristic
91	C₂²×C₄	Dic₅₈	1	2	yes	—
92	C₂²×C₄	D₁₁₆	1	2	yes	—
93	C₂²×C₄	D₁₁₆	1	2	yes	—
94	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
95	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
98	M₁₆	—	29	8	no	—
99	M₁₆	—	58	8	no	—
100	M₁₆	—	58	4	no	—
101	M₁₆	C₂×Dic₂₉	1	16	yes	—
102	C₂×C₈	—	58	8	no	—
103	C₂×C₈	—	58	4	no	—
104	C₂×C₈	C₂×Dic₂₉	1	4	yes	—
105	C₂×C₈	C₂×Dic₂₉	1	4	yes	—
106	C₂×C₈	C₂²×D₂₉	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₂₉	1	2	yes	—
109	C₂³	C₄⋊Dic₂₉	1	2	yes	—
110	C₂³	C₂×Dic₅₈	1	1	yes	characteristic
111	C₂³	C₂×Dic₅₈	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
113	C₂×C₄	C₄⋊Dic₂₉	1	4	yes	—
114	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₂₉	1	4	yes	—
116	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
117	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₁₆	1	4	yes	—
119	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
120	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
121	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
123	C₂×C₄	C₂²×Dic₂₉	1	4	yes	—
124	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
125	C₈	—	58	8	no	—
126	C₈	C₂²×Dic₂₉	1	8	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂×C₄⋊Dic₂₉	1	4	yes	—
132	C₂²	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
133	C₂²	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
134	C₂²	C₂²×Dic₅₈	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₂₉	1	4	yes	—
136	C₄	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
137	C₄	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
138	C₄	C₂²×D₁₁₆	1	1	yes	characteristic
139	C₄	C₂³×Dic₂₉	1	1	yes	characteristic
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#14271212	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₆
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂²×M₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₉	Q₈
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂²×M₁₆	D₂₉
hypocenter	C₂₉	C₂×#430
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #14271212	C₂	direct (N × H)
3. C₂×#11745167	C₂	semidirect
15. #11745167	C₂²	semidirect
75. C₂×C₂₉	#430	semidirect
88. C₂₉	C₂×#430	semidirect
143. C₂	#14271212	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#90777	2
#14271212	2
C₂×#11745167	2
C₂²×M₁₆×C₂₉	2
C₂×#430	29

Subgroups — C2×#14271212 #16828099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14271212 #16828099